Appendix

Chapter IV: Controlling light through complex scattering media with optical-

4.5 Appendix

Figure 4.6: Simulation results for comparison of the metrics, peak to background ratio (PBR) and contrast of noise ratio (CNR), which evaluate the quality of a focus pattern. (a) In optical wavefront shaping, at low PBR, e.g. PBR=2, the peak is immersed into the background of fully developed speckles, where the standard deviation of the speckle intensity is the same as its mean. In this case, CNR=PBR- 1=1. (b) For the same PBR, the time-averaged pattern created by OCIS shows a prominent peak as the variation of the background is much lower, resulting in a higher CNR, e.g. CNR=20 (∼1000 controllable modes). (c) To obtain the same CNR as the pattern formed by OCIS, the PBR of the focus formed by the wavefront shaping techniques needs to increase to 21. (d) The pattern in (b) is rescaled to help visually compare to the pattern in (c). As shown in (c) and (d), as long as the CNR is the same, the visibility of the peak is very similar although they have a very different PBR. Therefore, CNR is a more useful metrics for OCIS.

Figure 4.7: Experimental Setups. (a) Feedback based OCIS setup. (b) Optical intensity transpose setup, (b1) recording; (b2) playback. (c) Setup for direct imaging through scattering media. Abbreviations: AOM, acousto-optic modulator; BD, beam dump; BS, beam splitter; CAM, camera; DMD, digital micro-mirror device;

FB, fiber; G, ground glass diffuser; GM, galvanometer mirror; HWP, half wave plate; L, lens; PD, photodetector; PMT, photomultiplier tube; RD, rotating diffuser;

TP, target plane.

Figure 4.8: Experimental Setups. (a) Feedback based OCIS setup. (b) Optical intensity transpose setup, (b1) recording; (b2) playback. (c) Setup for direct imaging through scattering media. Abbreviations: AOM, acousto-optic modulator; BD, beam dump; BS, beam splitter; CAM, camera; DMD, digital micro-mirror device;

FB, fiber; G, ground glass diffuser; GM, galvanometer mirror; HWP, half wave plate; L, lens; PD, photodetector; PMT, photomultiplier tube; RD, rotating diffuser;

TP, target plane.

Figure 4.9: Experimental signal traces from the feedback-based OCIS. (a) A raw signal output from the photodetector during measurement. (b) A binarized signal output from the comparator during measurement. (c) A photodetector output signal during display. To provide a clearer visual comparison, we time-reversed this output signal again to match the timing.

Figure 4.10: CNR and PBR as a function of number of controllable modes. (a) CNR optimization mode based on Eq. 4.12 and Eq. 4.13 S8. (b) PBR optimization mode based on Eq. 4.14 and Eq. 4.15

Figure 4.11: Direct imaging through a thin scattering medium with OCIS. (a) Exper- imental setup. This procedure can be separated into two steps. (a1) Measurement.

This procedure is the same as the recording of the intensity reflection method de- scribed above. An optical spot was created on the target plane and a binarized speckle intensity is measured sequentially on the detector plane during a sweep of the galvo mirror. (a2) Direct observation. By using this signal to modulate the laser that illuminates a transmission object on the target plane, one can directly observe the object as the galvo mirror synchronizes with the modulated illumination. The method utilizes the angular memory effect of the thin scattering medium, where a tilted optical field incident to the thin scattering medium results in a tilted optical field on the other side. Therefore, the measured signal is also applicable to the neighboring points and enables direct observation of the object with only one mea- surement. (b) Optical diagram of the imaging process. (b1) Light from an object couples to different high-throughput channels over time and the transmitted light is directed to a spot to form a PSF of the imaging system. (b2) Based on the optical memory effect, a neighboring spot within the memory effect range also forms an im- age at the imaging plane. (c) Equivalently, OCIS and the scattering medium serves as an imaging system and one can see through the scattering medium directly. (d) An image of the object was formed through the scattering medium and captured by a camera on the detector plane. Scale bar: 10 𝜇m.

Figure 4.12: Optical information coupled out of the communication chain. (a) In free space, scatterers spread light to other directions. (b) In waveguide geometry, light can be coupled out of an optical fiber.

Supplementary notes Experimental setups

The optical setup of feedback-based OCIS is shown in Fig. 4.7a. A collimated CW laser beam (532 nm wavelength, CrystaLaser Inc.) was intensity-modulated by an acousto-optic modulator (AOM, 100MHz, IntraAction Corp.) by taking the first order of the diffracted beam. The modulated beam was then scanned by a galvo mirror (CRS 4 KHz, Cambridge Technology), which was imaged onto the surface of a ground glass diffuser (DG10-120, Thorlabs) through a 4-f relay system (L1, L2). The light intensity on the surface of the diffuser was 20 mW. Another 4-f system (L3, L4) magnified the speckle to match the core diameter of the fiber. A photomultiplier tube (PMT, H7422, Hamamatsu) was used to measure the speckle intensity, and the output signal was sent to an analog comparator (LM361N, Texas Instruments). An FPGA board (Cyclone 2, Altera) that was synchronized with the galvo mirror received and processed the output signals from the comparator. The output signals from the FPGA controlled an electronic switch (ZASWA-2-50DR+, Mini-circuits) to modulate the amplitude of the carrier (100 MHz) to the AOM. A camera (GX1920, Allied Vision) was placed on the conjugate plane of the fiber to observe the optical patterns.

The optical setup of optical intensity transpose is shown in Fig. 4.7b. During the recording process (Fig. 4.7b1), lens L3 created an optical spot behind the ground glass diffuser and the optical spot was conjugated with the camera by a 4-f system (L3 and L4). The light from the optical spot was then scattered by the diffuser and the PMT recorded the intensity on the Fourier plane of the galvo mirror which was conjugate to the surface of the diffuser. During the playback process (Fig. 4.7b2), the collimated laser beam that was aligned to be conjugated to the fiber end was modulated by the AOM when the galvo mirror was scanning. In the same way as feedback-based OCIS, the FPGA received the binarized signals from the comparator and output the signals to control the AOM for OCIS.

The optical setup for realizing imaging through scattering media is shown in Fig.

4.7c. We used a ground glass diffuser, the same as the one used in feedback- based OCIS demonstration, as the backscattering surface. The camera measured a speckle pattern after light backscattered from the surface. During data streaming, we randomly selected∼300 sub-channels (corresponding to∼300 speckles) to form a channel. The light intensity modulation was realized with a DMD system (Discovery 4100, Texas Instruments). To assure linear intensity operation as described in the

intensity transmission matrix theory, the DMD modulates spatially incoherent light, which was scattered by a rotating diffuser in front of the coherent laser source.

Mathematical derivation of CNR and PBR OCIS optical spot

Here we quantitatively evaluate the performance of OCIS techniques. Assuming the instantaneous speckle patterns are fully developed [24], the speckle intensity follows an exponential distribution with mean 𝜇and standard deviation 𝜎 = 𝜇. The shot noise effect will be considered in the next section. The probability density function is given by

𝑃(𝐼) = 1 𝜇

exp(−𝐼 𝜇

), (4.6)

where𝜇is the mean intensity of the speckle pattern. Then,𝛼, the portion of patterns in which the intensity value of the pixel of interest is higher than a threshold 𝐼_𝑡 is given by

𝛼=

∫ ∞

𝐼_𝑡

1 𝜇

exp(−𝐼 𝜇

)𝑑𝐼 =exp−𝐼 𝜇

. (4.7)

The mean intensity of the pixel of interest among these patterns is therefore given by

¯ 𝐼_𝑝 =

∫ ∞

𝐼_𝑡

𝐼× 𝑃(𝐼) 𝛼

𝑑𝐼 . (4.8)

Substituting Eq. 4.6 and 4.7 into Eq. 4.8 leads to

¯

𝐼_𝑝 =𝜇+𝐼_𝑡. (4.9)

If our system captures𝑁independent speckle patterns in total, the number of selected patterns is then approximately𝛼 𝑁. Since the OCIS sums up all the selected patterns, the peak intensity of the resultant pattern on average is given by ¯𝐼_{𝑠 𝑝} =𝛼 𝑁𝐼¯_𝑝, while the mean and standard deviation of the background of the resultant pattern is given by 𝐼_{𝑠 𝑏} = 𝛼 𝑁 𝜇 and 𝜎_{𝑠 𝑏} = √

𝛼 𝑁 𝜎 = √

𝛼 𝑁 𝜇. The contrast-to-noise ratio (CNR) is given by

𝐶 𝑁 𝑅_𝑃 =

¯

𝐼_{𝑠 𝑝}−𝐼¯_{𝑠 𝑏} 𝜎_{𝑠 𝑏}

=

√ 𝛼 𝑁

𝐼_𝑡 𝜇

=

√

𝑁exp(− 𝐼_𝑡 2𝜇

)𝐼_𝑡 𝜇

. (4.10)

The PBR of OCIS is given by 𝑃 𝐵 𝑅_𝑃 =

¯ 𝐼_{𝑠 𝑝}

¯ 𝐼_{𝑠 𝑏}

= 𝛼 𝑁𝐼¯_𝑝

𝛼 𝑁 𝜇

= 𝜇+𝐼_𝑡 𝜇

=1+ 𝐼_𝑡 𝜇

. (4.11)

From Eq. 4.10 and 4.11, we find that both CNR and PBR are functions of intensity threshold 𝐼_𝑡 that we choose. Therefore, by selecting a proper threshold, we can

optimize the CNR or PBR accordingly. Here we analyze the solutions for CNR and PBR optimization, respectively. The subscripts “A” and “B” in CNRs and PBRs below correspond to “a. CNR optimization” and “b. PBR optimization”, respectively.

a. CNR optimization

Through optimization, we find that maximum CNR is achieved when the intensity threshold is set at double of the mean intensity, that is 𝐼_𝑡 = 2𝜇. In this case, the CNR given by Eq. 4.10 becomes

𝐶 𝑁 𝑅_{𝑃_𝐴} = 2 𝑒

√

𝑁 (4.12)

and the PBR given by Eq. 4.11 becomes

𝑃 𝐵 𝑅_{𝑃_𝐴} =3. (4.13)

In this case, PBR decouples from CNR and is a constant independent of the number of summed speckle patterns. In our experiment (Fig. 4.3b in the article, 𝜏= 500 ms), we achieved a PBR of∼2.5.

b. PBR optimization

To maximize PBR, one would set 𝐼_𝑡 as high as possible as indicated by Eq. 4.11.

However, the maximum 𝐼_𝑡 is bounded by the requirement that on average one speckle pattern is selected during display. This requirement can be describes as 𝛼 𝑁 = 1. Substituting𝛼with Eq. 4.7, we find that the intensity threshold for PBR optimization is given by 𝐼_𝑡 = 𝜇ln𝑁 . Substituting this equation into Eq. 4.11, we have the maximum PBR:

𝑃 𝐵 𝑅_{𝑃_𝐵} =1+ln𝑁 . (4.14)

Using this intensity threshold to calculate CNR based on Eq. 4.10, we have

𝐶 𝑁 𝑅_{𝑃_𝐵} =ln𝑁 . (4.15)

Equations 4.14 and 4.15 show that the PBR and CNR are coupled in this case. This relationship, 𝑃 𝐵 𝑅 = 𝐶 𝑁 𝑅 +1, is the same as that in wavefront shaping because in both cases the intensity distribution of the background follows speckle intensity distribution where its mean intensity equals its standard deviation.

Figure 4.10 plots CNR and PBR as a function of number of measured optical modes.

From this figure, we can find that in CNR optimization mode, CNR increases as

a function of number of optical modes while the PBR remains the same. The enhancement in CNR means that the optical spot is more evident, which is the key metric to evaluate the ability of an imaging technique. In contrast, PBR fails to indicate this ability in this mode as it remains constant over the number of optical modes. In the PBR optimization mode, PBR and CNR are coupled and thus both of them can be used to evaluate the performance of an imaging technique.

Null energy point

For an OCIS generated null energy point, the portion of speckle patterns being selected is given by

𝛼=

∫ ^𝐼𝑡

0

1

𝜇exp−𝐼 𝜇

𝑑𝐼 =1−exp(−𝐼_𝑡 𝜇

). (4.16)

In this case, the expected intensity of the point of interest among the selected patterns is given by

¯ 𝐼_𝑛 = 1

𝛼

∫ 𝐼𝑡

0

𝐼 ×𝑃(𝐼)𝑑𝐼 . (4.17)

By substituting the probability density function𝑃(𝐼)with Eq. 4.6, we have

¯

𝐼_𝑛 =𝐼_𝑡 − 𝐼_𝑡 1−exp(−^𝐼_𝜇^𝑡)

+𝜇. (4.18)

If the system measures 𝑁 speckle patterns in total, the number of selected speckle patterns is𝛼 𝑁, and therefore the expected intensity of the sum of these patterns at the null energy point is given by 𝐼_𝑠𝑛 = 𝛼 𝑁𝐼¯_𝑛. Likewise, the expected intensity of the sum of these patterns at the background is given by𝐼_{𝑠 𝑏} =𝛼 𝑁 𝜇, and the standard deviation of the background is𝜎_{𝑠 𝑏} =√

𝛼 𝑁 𝜎=√

𝛼 𝑁 𝜇. Therefore, the CNR, which is defined as the ratio between the background-subtracted null intensity and the standard deviation of the background, is given by

𝐶 𝑁 𝑅_𝑁 = 𝐼¯_𝑠𝑛−𝐼¯_{𝑠 𝑏} 𝜎_{𝑠 𝑏}

=−

√ 𝑁

𝐼_𝑡 𝜇

exp(−^𝐼_𝜇^𝑡) 1−exp(−^𝐼_𝜇^𝑡)

. (4.19)

The PBR, which is defined as the ratio between the negative peak or null point intensity and the mean of the background, is given by

𝑃 𝐵 𝑅_𝑁 = 𝐼¯_𝑠𝑛

¯ 𝐼_{𝑠 𝑏}

=−√ 𝑁

𝐼_𝑡 𝜇

exp(−^𝐼𝑡

𝜇) 1−exp(−^𝐼_𝜇^𝑡)

. (4.20)

Discussions on the security of OCIS-based communications

There are two typical scenarios where light can be received by a third party (Fig.

4.12). In free space, scatterers, such as dust, fog, turbid water, or opaque walls, scatter light outside the line-of-sight of the communication parties. In waveguide geometry, leaky modes allow the light to be coupled out of the waveguide. There is also an extreme case where an optical fiber waveguide is cut and a beam splitter is inserted in between. Although this behavior can be easily monitored, we also include it in our security analysis framework.

Without OCIS, light scattering and coupling into a third party will allow the third party to receive the same copy of the information as the primary communication parties. In this case, the security of the information only depends on the use of a digital key to encrypt the information. If the third party hacks the digital key, the information is revealed.

OCIS provides a physical layer of encryption, which can be used on top of digi- tal encryption. Here we analyze the probability of decoding the OCIS encrypted information by coupling and detecting the light during propagation in the aforemen- tioned scenarios. In principle, if the third party (Chuck) can measure the full optical field from the primary communication parties (Alice and Bob), he can decode the information by correlating the two optical fields based on the time-reversal symme- try of light propagation. In practice, measuring the optical field in the middle of the scattering media is extremely challenging in OCIS for several reasons. First, measuring the full field requires a full coverage in free space or cutting the optical fibers, which can be easily monitored as discussed above. Second, OCIS can use multiple spatially incoherent light sources, between which there is no static phase difference, to prevent phase measurement. Therefore, we would like to focus on a more practical case where intensity patterns are measured in the middle.

Here is the process of the intensity pattern measurement. First, Alice sends a single- mode laser pulse through the scattering media to establish a channel map with Bob.

Chuck measures a speckle pattern in the middle of the scattering medium, and Bob measures a speckle pattern on the other end of the scattering medium. For simplicity, here we analyze the case where Bob only sends light through one channel for one bit of information transmission. This channel is randomly selected from the channels that meet the intensity requirement and the scattering medium is refreshed when all the channels have been used. Chuck measures the second speckle pattern in the middle when Bob sends one bit back to Alice. In this case, Chuck tries to decode

the information by calculating the sign of the correlation coefficient between the speckle patterns.

Mathematically, we can explicitly calculate the correlation coefficient 𝐶 of the intensity patterns measured by Chuck and analyze its expected value and the standard deviation. The correlation coefficient𝐶 has the form of

𝐶 =

1 𝑀₀

Í^𝑀0

𝑟=1(𝐼_{𝐶 , 𝐴}(𝑟) −𝐼¯_{𝐶 , 𝐴}) (𝐼_{𝐶 , 𝐵}(𝑟) −𝐼¯_{𝐶 , 𝐵})

¯ 𝐼_{𝐶 , 𝐴}𝐼¯_{𝐶 , 𝐵}

, (4.21)

where 𝐼_{𝐶 , 𝐴}(𝑟) and 𝐼_{𝐶 , 𝐵}(𝑟) are the intensity patterns measured by Chuck when Alice and Bob send the light pulses, respectively; ¯𝐼_{𝐶 , 𝐴} and ¯𝐼_{𝐶 , 𝐵} are the mean intensities of these two patterns, respectively; M0 is the total number of spatial modes generated by the scattering medium and is much larger than one; 𝑟 is the index of the speckle grains. After mathematical derivation based on the complex field relationship ensured by reciprocity, the expected value of correlation coefficient 𝐶has the following expression:

𝐸(𝐶) ≈ 𝐼_𝑡−𝐼¯ 𝑀₀𝐼¯

, (4.22)

where𝐼_𝑡 is the intensity of the speckle grain that Bob selects as the channel to send one bit back to Alice; ¯𝐼is the mean intensity of the speckle grains at Bob’s side. For simplicity, here we assume that Alice and Bob use the same amount of power for the laser pulses they send to each other. In this case, the speckle power that Alice observes is also 𝐼_𝑡, the same as that of the speckle that Bob selects based on the intensity transmission matrix theory.

While the step by step derivation of the correlation coefficient in Eq. 4.22 is beyond the scope of the work, this equation has an intuitive interpretation. The numerator 𝐼_𝑡 − 𝐼¯indicates the power deviated from the mean power at the mode of interest that Alice observes or Bob selects. If Bob randomly picks a channel to send light back to Alice, the expected value of this deviation should be zero, and the expected correlation between Chuck’s patterns is also zeros. Therefore, the expected value of the correlation coefficient describes the energy ratio between the part that is deviated from the mean at the mode of interest and the total energy.

For each bit during transmission, Chuck does the correlation between the two speckle patterns and obtains one correlation coefficient 𝐶. Therefore, it is also important to know the deviation of the one-time calculation from the expected value of the correlation coefficient 𝐶. The error or the standard deviation of the correlation

coefficient is given by [25]

𝑠𝑡 𝑑(𝐶) ≈ r 1

𝑀

, (4.23)

where𝑀is the number of modes that Chuck measures out of the𝑀₀modes carried by the scattering medium. Here we assume that the measurement is well above shot noise limit. Therefore, the SNR of the information that Chuck obtained is given by

𝑆 𝑁 𝑅_𝐶 = ( 𝐸(𝐶)

𝑠𝑡 𝑑(𝐶))²= (𝐼_𝑡− 𝐼¯

¯ 𝐼

)² 𝑀 𝑀²

0

. (4.24)

Here we provide an example calculation of the SNR that Chuck may receive. Let’s assume that a scattering medium carries 10⁶ modes (𝑀₀) and Chuck measures all the modes in an extreme case (𝑀 = 𝑀₀); the mean of the threshold that Bob chooses is 2 ¯𝐼. In this case, the SNR of the correlation coefficient 𝐶 is ∼ 10⁻⁶, which is very difficult for Chuck to obtain meaning information. In practice, Chuck can only measure a small portion of the modes, resulting in an even lower SNR. The leakage of information can be further mitigated by the combination of digital encryption, such as leakage-resilient cryptography [26].

By providing a physical layer of encryption, OCIS based secure communication can be potentially applied to several scenarios including free-space and fiber-based communication. Importantly, this physical encryption is complementary to and able to work with key based encryption, which includes keys that are generated with optical approaches such as quantum key distribution [27]. Compared to quantum key distribution, OCIS does not have a strict requirement on the number of photons used in communications as long as Alice and Bob can measure sufficient photons.

It should be noted that OCIS requires multimode fibers to provide the physical encryption, which is likely to be a limiting factor for immediate use in some existing networks that are based on single mode fibers. In the demonstration, we only show a one-way communication where Bob sends information to Alice. Extending to a two- way communication is straightforward - Alice will need access to multiple speckles like Bob. In our experiment, the data transfer rate is limited by the refreshing rate of the DMD. The data transfer rate can possibly be improved by using an acousto-optic deflector (AOD) to select the intensity channels in the future.

Supplementary Methods - Image transmission through scattering media with OCIS

With the knowledge of intensity mapping between the input plane and target plane, OCIS is able to correct for disordered scattering and allows for direct transmission

of intensity information through scattering media. Here, we demonstrate this ability by directly imaging an object through a scattering medium. From the recording process of optical intensity transpose (Fig. 4b1 or Fig. 4.11a1), we can obtain a map of optical channels between the input plane and target plane during a galvo mirror scan. We can then direct the light from the high-throughput channels to a point on the detector plane during the second galvo scan. In this case, we modulate and send light to the high throughput channels sequentially when the galvo mirror rotates to positions where the channels are connected to the point (Fig. 4.11a1). As such, we obtain a time-averaged optical spot on the detector plane as a PSF of the imaging system.

To form a wide field image through the scattering medium, here we utilize the tilt-tilt correlation or angular memory effect of a thin scattering medium [28, 29]. Within an angular memory effect range, tilting of an input wavefront to a scattering medium causes tilting of the scattered output wavefront, and these two optical wavefronts remain highly correlated. For a thin scattering medium, the correlation maintains within a reasonable tilting angle for wide field imaging. Therefore, the modulation signal that generates the PSF is also a valid solution to cast a neighboring spot on the target plane to a shifted optical spot on the detector plane through the scattering medium (Fig. 4.11a2 and 4.11b2). The method maps to the phase compensation approach that enables wide field imaging through thin scattering medium in optical wavefront shaping [29, 30]. In both cases, we can interpret the system as a piece of compensation optics that corrects for the scattering of the sample and allows us to see through the scattering medium directly (Fig. 4.11c). Intriguingly, no phase information or manipulation is required for OCIS to compensate for the optical scattering here.

To directly correct optical scattering and form an image in free space through a thin scattering medium experimentally, we first calibrated the scattering medium by measuring the response of a point source on the target plane through the scattering medium (Fig. 4.11a1). See Fig. 4.7d for more details on the setup. This step is the same as the recording process in optical intensity transpose. We then used a target consisting of two points near the calibration point with a separation of 20 𝜇m (Fig. 4.11a2). To image the object, the laser source was modulated with the signal measured from the calibration step as the galvo mirror scans. We placed a camera with an exposure time covering the galvo scan duration to directly observe the image of the two spots on the detector plane. As shown in Fig. 4.11d, the image